91Ó°ÊÓ

Find the stable, unstable and center subspaces \(E^{a}, E^{u}\) and \(E^{c}\) of the linear system (1) with the matrix (a) \(A=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\) (b) \(A=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]\) (c) \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) (d) \(A=\left[\begin{array}{rr}-1 & -3 \\ 0 & 2\end{array}\right]\) (e) \(A=\left[\begin{array}{rr}2 & 3 \\ 0 & -1\end{array}\right]\) (f) \(A=\left[\begin{array}{rr}2 & 4 \\ 0 & -2\end{array}\right]\) (g) \(A=\left[\begin{array}{rr}0 & 0 \\ 0 & -1\end{array}\right]\) (h) \(A=\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right]\) (i) \(A=\left[\begin{array}{rr}-1 & -1 \\ 1 & -1\end{array}\right]\) Also, sketch the phase portrait in each of these cases. Which of these matrices define a hyperbolic flow, \(e^{A t}\) ?

Find the Jordan canonical forms for the following matrices (a) \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) (b) \(A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\) (c) \(A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\) (d) \(A=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\) (e) \(A=\left[\begin{array}{rr}1 & 1 \\ 0 & -1\end{array}\right]\) (f) \(A=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]\) (g) \(A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\) (h) \(A=\left[\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right]\) (i) \(A=\left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]\).

Compute the operator norm of the linear transformation defined by the following matrices: (a) \(\left[\begin{array}{rr}2 & 0 \\ 0 & -3\end{array}\right]\) (b) \(\left[\begin{array}{rr}1 & 2 \\ 0 & -1\end{array}\right]\) (c) \(\left[\begin{array}{ll}1 & 0 \\ 5 & 1\end{array}\right]\). Hint: In (c) maximize \(|A x|^{2}=26 x_{1}^{2}+10 x_{1} x_{2}+x_{2}^{2}\) subject to the constraint \(x_{1}^{2}+x_{2}^{2}=1\) and use the result of Problem 2 ; or use the fact that \(\|A\|=\left[\text { Max eigenvalue of } A^{T} A\right]^{1 / 2}\). Follow this same hint for \((b)\).

Show that the operator norm of a linear transformation \(T\) on \(\mathbf{R}^{n}\) satisfies $$ \|T\|=\max _{|\mathbf{X}|=1}|T(\mathbf{x})|=\sup _{\mathbf{x} \neq 0} \frac{|T(\mathbf{x})|}{|\mathbf{x}|} $$

Solve the linear system \(\dot{x}=A x\) and sketch the phase portrait for (a) \(A=\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]\) (c) \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & 3\end{array}\right]\). (b) \(A=\left[\begin{array}{ll}3 & 0 \\ 0 & 1\end{array}\right]\) (d) \(A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\).

If \(\operatorname{det} A=0\), then the origin is a degenerate critical point of \(\dot{x}=A x\). Determine the solution and the corresponding phase portraits for the linear system with (a) \(A=\left[\begin{array}{ll}\lambda & 0 \\ 0 & 0\end{array}\right]\) (b) \(A=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]\) (c) \(A=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\) Note that the origin is not an isolated equilibrium point in these cases. The four different phase portraits determined in (a) with \(\lambda>0\) or \(\lambda<0\), (b) and (c) above, together with the sources, sinks, centers and saddles discussed in this section, illustrate the eight different types of qualitative behavior that are possible for a linear system.

Describe the separatrices for the linear system $$ \begin{aligned} &\dot{x}_{1}=x_{1}+2 x_{2} \\ &\dot{x}_{2}=3 x_{1}+4 x_{2} \end{aligned} $$ Hint: Find the eigenspaces for \(A\).

Let the \(2 \times 2\) matrix \(A\) have real, distinct eigenvalues \(\lambda\) and \(\mu\). Suppose that an eigenvector of \(\lambda\) is \((1,0)^{T}\) and an eigenvector of \(\mu\) is \((-1,1)^{T}\). Sketch the phase portraits of \(\dot{x}=A x\) for the following cases: (a) \(0<\lambda<\mu\) (b) \(0<\mu<\lambda\) (c) \(\lambda<\mu<0\) (d) \(\lambda<0<\mu\) (e) \(\mu<0<\lambda\) (f) \(\lambda=0, \mu>0\).